Finite-temperature semiclassical expansion of the one-body density matrix and kinetic-energy density in d dimensions

We develop a finite-temperature semiclassical expansion for the one-body density matrix and the associated kinetic-energy density of noninteracting fermions with constant mass in arbitrary spatial dimension d. The analysis is performed analytically up to order ħ2 and relies on a generalized class of Fermi–Bessel integrals, which allows us to obtain fully closed-form expressions for all gradient corrections. The resulting formulation provides a continuous interpolation between the quantum-degenerate limit and the classical Maxwell–Boltzmann regime. As T→0, the expansion consistently reduces to the Thomas–Fermi term together with Weizsäcker-type gradient corrections, while at finite temperature it yields explicit gradient contributions that were previously unavailable in closed form for generic dimension d. The framework presented here establishes a compact and dimension-independent analytical basis for semiclassical modeling of inhomogeneous fermionic matter and enables systematic finite-temperature extensions of orbital-free density functional theory.